Newbie question about extrusion of hemispheres
bottom left:
Create a sphere and a cube. 
Set a Boolean modifier on the cube.
Set the modifier's operation to "difference" and use the sphere for its object. 

bottom right:
this is the result

top left:
Subdivide a plane or mesh 2 times
Inset the 4 center faces
Extrude
You could at this point select the extrusion and hit alt shift s to spherize the center. Otherwise, just try to make the shape I showed up there by hitting shift+z while scaling the center most verts. 

top right:
this is a subdivide modifier applied to the top left mesh